Methods Of Statistical Mechanics Research Articles

Synthetic first-principles studies from phase equilibria to microstructural formation in the Fe-Pt L10 phase

Electronic-structure calculations, the cluster-variation method of statistical mechanics, and the phase-field method were combined in attempted first-principles calculations of phase equilibria and microstructural evolution associated with the disorder-$\mathrm{L}{1}_{0}$ transition of the Fe-Pt system. The calculated disorder-$\mathrm{L}{1}_{0}$ transition temperature was within $\ensuremath{\sim}10$ K difference from the experimental value, and the locus of spinodal ordering temperature is placed in the phase diagram. The calculated microstructure demonstrates preferential growth of the ordered domain along the 100> direction and, in the later period, an anisotropic morphology of an antiphase domain structure develops. We offered an interpretation from the atomistic point of view for this morphology. We therefore achieved consistent first-principles multiscale calculations of phase equilibria and microstructural evolution, bridging microscopic to mesoscopic scales without any adjusting parameters.

Computing chemical potentials of adsorbed or confined fluids.

The chemical potential of adsorbed or confined fluids provides insight into their unique thermodynamic properties and determines adsorption isotherms. However, it is often difficult to compute this quantity from atomistic simulations using existing statistical mechanical methods. We introduce a computational framework that utilizes static structure factors, thermodynamic integration, and free energy perturbation for calculating the absolute chemical potential of fluids. For demonstration, we apply the method to compute the adsorption isotherms of carbon dioxide in a metal-organic framework and water in carbon nanotubes.

Resurrection of Hückel’s idea: Decoupling ion–ion and ion–water terms in activity coefficients via the state-dependent dielectric constant

Although Hückel proposed the basic idea of using a concentration dependent dielectric constant, ϵ(c), to compute the activity coefficients of ions in electrolytes in 1925 (Hückel, 1925), a large amount of modeling studies appeared in the literature only after 2010 when we published our II+IW theory (Vincze et al., 2010) that splits the excess chemical potential into two terms corresponding to interactions between ions (II) and interactions between ions and water (IW). In this approach, the two terms are decoupled, which means that they can be computed independently with ϵ(c) being the only link between them. Here, we review our theory and other works based on Hückel’s suggestion by discussing several issues that are partly cornerstones of the theory, partly make it possible to put the theory into a larger context. These issues include the role of ϵ(c) and the ionic radii used in the II and IW term, the statistical mechanical methods to estimate the II term, the existence and interpretation of individual activities, phenomena associated with strong ionic correlations in multivalent electrolytes, and explicit-water models used in molecular dynamics simulations.

Effect of Coupling Discretization on Coherent-Ising-Machine-Implemented Hopfield Model

The coherent Ising machine (CIM) is being developed for solving combinatorial optimization problems scalably at high speed. The actual equipment of the measurement-feedback type of CIM (MFB-CIM) uses a field-programmable gate array (FPGA) to calculate the local field. Because the product of the coupling strength matrix and the measured amplitude vector of the optical parametric oscillator pulses must be calculated in real-time to achieve feedback, the coupling strength must have a low-bit discrete representation in the FPGA. By reducing the number of bits of coupling strength, the amount of memory needed to store the coupling matrix can be reduced and the calculation speed increased so that the MFB-CIM can implement more spins. Therefore, it is necessary to reduce the number of bits of coupling strength as much as possible while maintaining the performance in solving combinatorial optimization problems. In this paper, to evaluate the effect of discretization of the coupling strength in the MFB-CIM, we quantitatively evaluate the performance of the Hopfield model with discrete coupling when it is implemented in the CIM. The Hopfield model, which is a canonical model of Ising computation in information statistical mechanics, is defined in a coupling manner called the correlation-type interaction, in common with other models such as Sourlas code, a code division multiple access multiuser detector, and L0 regularization-based compressed sensing. We derived the macroscopic equations (MEs) of the Hopfield model with discrete coupling when it is implemented in the CIM by using the non-equilibrium statistical mechanics method we previously proposed. Moreover, we clarified the relationship between the number of bits of the coupling strength and the critical memory capacity for various pump rates.

Diverse phase transitions in optimized directed network models with distinct inward and outward node weights.

We consider growing directed network models that aim at minimizing the weighted connection expenses while at the same time favoring other important network properties such as weighted local node degrees. We employed statistical mechanics methods to study the growth of directed networks under the principle of optimizing some objective function. By mapping the system to an Ising spin model, analytic results are derived for two such models, exhibiting diverse and interesting phase transition behaviors for general edge weight, inward and outward node weight distributions. In addition, the unexplored cases of negative node weights are also investigated. Analytic results for the phase diagrams are derived showing even richer phase transition behavior, such as first-order transition due to symmetry, second-order transitions with possible reentrance, and hybrid phase transitions. We further extend previously developed zero-temperature simulation algorithm for undirected networks to the present directed case and for negative node weights, and we can obtain the minimal cost connection configuration efficiently. All the theoretical results are explicitly verified by simulations. Possible applications and implications are also discussed.

Phase transition in compressed sensing with horseshoe prior.

In Bayesian statistics, horseshoe prior has attracted increasing attention as an approach to compressed sensing. By considering compressed sensing as a randomly correlated many-body problem, statistical mechanics methods can be used to analyze the problem. In this paper, the estimation accuracy of compressed sensing with the horseshoe prior is evaluated by the statistical mechanical methods of random systems. It is found that there exists a phase transition in signal recoverability in the plane of the number of observations and the number of nonzero signals, and that the recoverable phase is more extended than that using the well-known l_{1} norm regularization.

Statistical Mechanical Analysis of Phase Unwrapping —One-Dimensional Model—

We study a phase unwrapping model in the one-dimensional space on the basis of Bayesian inference using the maximizer of posterior marginals (MPM) estimate by the statistical mechanical methods. We propose a model in which the recursion relations to obtain statistical quantities such as MPM estimates are derived. We introduce the three state Potts model to handle the discontinuities in observed data, and propose two methods, the step and direct methods. We derive the recursion relations for MPM estimates of hyperparameters and phase differences in both methods, and investigate the random and regular phase differences, and previously studied other type of random phase differences. We find that the phase differences are inferred fairly well in rather wide ranges of noise amplitudes. The ranges depend on samples and the system sizes. Furthermore, we find that the step method has performance in phase unwrapping comparable to the direct method, and that it is much faster in numerical computation and applicable to much larger system sizes than the direct method.

Using the Monte-Carlo Method with Subsequent Approximation in Statistical-Mechanical Modeling for Estimating the Reliability of NPP Pipelines and Pressure Vessels

For a large number of safety-critical components at existing and newly designed NPPs, the task of confirming their reliability by conventional statistical methods does not seem to be possible in view of both short periods of time for which these components have been in operation and almost no failures that have occurred for this period of time, as well as lack of analogs from the worldwide practices. One of approaches to assessing the failure probabilities of highly reliable reactor plant equipment involves application of the statistical-mechanical modeling method. For solving statistical-mechanical models, the Monte Carlo (MC) method, which has a number of advantages over analytical ones, is commonly used. However, the need to perform an excessively large number of statistical simulations in substantiating low event probabilities prompts specialists to search for alternative calculation methods. One of these techniques, proposed in this article, is a two-stage method that combines the Monte Carlo method and approximation of statistical data – the Monte Carlo method with subsequent approximation. Matters of reliability assessment with using statistical-mechanical modeling of components’ limit states are considered. The use of the proposed Monte Carlo method with subsequent approximation makes it possible to reduce the number of statistical simulations while preserving the accuracy of the results. This statement has been confirmed by a number of examples of calculating the probabilities of leaks and ductile-brittle fracture of reactor plant components.

A kinetic description of the body size distribution of species

In this paper, by resorting to classical methods of statistical mechanics, we build a kinetic model able to reproduce the observed statistical weight distribution of many diverse species. The kinetic description of the time variations of the weight distribution is based on elementary interactions that describe in a qualitative and quantitative way successive evolutionary updates, and determine explicit equilibrium distributions. Numerical fittings on mammalian eutherians of the order Chiroptera population illustrates the effectiveness of the approach.

Phosphorylation of Thr9 Affects the Folding Landscape of the N-Terminal Segment of Human AGT Enhancing Protein Aggregation of Disease-Causing Mutants.

The mutations G170R and I244T are the most common disease cause in primary hyperoxaluria type I (PH1). These mutations cause the misfolding of the AGT protein in the minor allele AGT-LM that contains the P11L polymorphism, which may affect the folding of the N-terminal segment (NTT-AGT). The NTT-AGT is phosphorylated at T9, although the role of this event in PH1 is unknown. In this work, phosphorylation of T9 was mimicked by introducing the T9E mutation in the NTT-AGT peptide and the full-length protein. The NTT-AGT conformational landscape was studied by circular dichroism, NMR, and statistical mechanical methods. Functional and stability effects on the full-length AGT protein were characterized by spectroscopic methods. The T9E and P11L mutations together reshaped the conformational landscape of the isolated NTT-AGT peptide by stabilizing ordered conformations. In the context of the full-length AGT protein, the T9E mutation had no effect on the overall AGT function or conformation, but enhanced aggregation of the minor allele (LM) protein and synergized with the mutations G170R and I244T. Our findings indicate that phosphorylation of T9 may affect the conformation of the NTT-AGT and synergize with PH1-causing mutations to promote aggregation in a genotype-specific manner. Phosphorylation should be considered a novel regulatory mechanism in PH1 pathogenesis.

Reliable emulation of complex functionals by active learning with error control.

A statistical emulator can be used as a surrogate of complex physics-based calculations to drastically reduce the computational cost. Its successful implementation hinges on an accurate representation of the nonlinear response surface with a high-dimensional input space. Conventional "space-filling" designs, including random sampling and Latin hypercube sampling, become inefficient as the dimensionality of the input variables increases, and the predictive accuracy of the emulator can degrade substantially for a test input distant from the training input set. To address this fundamental challenge, we develop a reliable emulator for predicting complex functionals by active learning with error control (ALEC). The algorithm is applicable to infinite-dimensional mapping with high-fidelity predictions and a controlled predictive error. The computational efficiency has been demonstrated by emulating the classical density functional theory (cDFT) calculations, a statistical-mechanical method widely used in modeling the equilibrium properties of complex molecular systems. We show that ALEC is much more accurate than conventional emulators based on the Gaussian processes with "space-filling" designs and alternative active learning methods. In addition, it is computationally more efficient than direct cDFT calculations. ALEC can be a reliable building block for emulating expensive functionals owing to its minimal computational cost, controllable predictive error, and fully automatic features.

On the Correspondence between Subshifts of Finite Type and Statistical Mechanics Models.

Several classical problems in symbolic dynamics concern the characterization of the simplex of measures of maximal entropy. For subshifts of finite type in higher dimensions, methods of statistical mechanics are ideal for dealing with these problems. R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics-a correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it and present some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way the possibilities of this correspondence.

Heterogeneous consecutive reaction kinetics of direct oxidation of H2 to H2O2: Effect and regulation of confined mass transfer

Porous catalysts in heterogeneous reactions have played an important role in the modern chemical industry, but it is still challenging to quantitatively describe mass transfer and surface reaction behaviors of reactants in nano-confined space. Direct synthesis of hydrogen peroxide (H2O2) is considered as an attractive alternative to anthraquinone oxidation process, while the confined mass transfer of H2O2 in porous catalysts limits the reactivity. In this work, taking the consecutive reaction of H2O2 synthesis as an example, a quantitative method in modeling the effects of confined mass transfer on the reactivity was studied. More specifically, calorimetry was developed to characterize the confined structures of porous carbon experimentally, the linear nonequilibrium thermodynamics and the statistical mechanics method were further combined. Then, the heterogeneous consecutive reaction kinetics and the Thiele modulus influenced by confined mass transfer were modeled. Consequently, regulation strategies were proposed with the help of theoretical models. The optimized catalyst with biological skeleton carbon support and 0.5 wt% palladium loading shows an excellent catalytic performance. Lastly, for the mesoscience in heterogeneous reaction, the resistance was explored as a quantitative descriptor to compromise in the competition between mass transfer and surface reaction. The mesoscale structures were considered as the dynamic spatiotemporal distribution of substance concentrations, and the resistance minimization multi-scale (RMMS) model was proposed.

Second- to first-order phase transition: Coevolutionary versus structural balance

In social networks, the balance theory has been studied by considering either the triple interactions between the links (structural balance) or the triple interaction of nodes and links (coevolutionary balance). In the structural balance theory, the links are not independent from each other, implying a global effect of this term and it leads to a discontinuous phase transition in the system's balanced states as a function of temperature. However, in the coevolutionary balance the links only connect two local nodes and a continuous phase transition emerges. In this paper, we consider a combination of both to understand which of these types of interactions will identify the stability of the network. We are interested to see how adjusting the robustness of each term versus the other might affect the system to reach a balanced state. We use statistical mechanics methods and the mean-field theory and also the Monte Carlo numerical simulations to investigate the behavior of the order parameters and the total energy of the system. We find the phase diagram of the system which demonstrates the competition of these two terms at different ratios against each other and different temperatures. The system shows a tricritical point above which the phase transition switches from continuous to discrete. Also the superiority of the local perspective is observed at low temperatures and the global view will be the dominant term in determining the stability of the system at higher temperatures.

Mean-field theory of the spin-1/2 transverse field Ising model with a negative thermal expansion

A generalized spin-1/2 transverse field Ising model with a negative thermal expansion of the lattice is introduced and investigated using standard methods of statistical mechanics. Besides the volume-dependent magnetic energy, the static lattice energy, and anharmonic Einstein phonon energy are also considered in calculations. Analytic relations for the Gibbs free energy, magnetic moments, and equations of state are obtained, taking into account a simple volume dependence of all energy contributions. The ground-state and finite-temperature phase diagrams are discussed in detail for the strong and weak magneto-elastic coupling. It is clearly demonstrated that the generalized spin-1/2 transverse field Ising model exhibits a novel critical behavior, due to the strong negative expansion of the lattice, which is controlled by the strength of magneto-elastic coupling. The presented approach can be easily modified to study also other magnetic and non-magnetic crystalline models of solids.

Revisiting the melting curves of vanadium and niobium metals under pressure

The Lindemann criterion is adjusted to revisit melting curves of vanadium and niobium metals under pressure thanks to the statistical moment method (SMM) in statistical mechanics. The equation-of-state and atomic mean-square displacement in SMM approach are applied to implement numerical calculations for vanadium and niobium metals up to 250 GPa and 150 GPa, respectively. The obtained theoretical melting curves are compared with results of previous experiments and molecular dynamics (MD) simulations where possible to check the theory. We expose that our theoretical melting line of vanadium is overall in good agreement with previous results determined by in situ X-ray diffraction (XRD) and diffuse scattering signals using synchrotron XRD together with shock experiments up to pressure 250GPa. For niobium, our melting curve follows very well with those of recent experimental XRD data as well as ab initio Z method up to 124 GPa. However, we also observe the difference of classical MD simulation results from SMM calculations at pressure below 2.5 GPa. This research enriches the high-pressure melting point database as well as provides a relatively simple approach with low computational cost to estimate high-pressure melting points of materials.

Macroscopic Analysis of Vector Approximate Message Passing in a Model-Mismatched Setting

In this study, macroscopic properties of the vector approximate message passing (VAMP) algorithm for inference of generalized linear models are investigated using a non-rigorous heuristic method of statistical mechanics when the true posterior cannot be used and the measurement matrix is a sample from rotation-invariant random matrix ensembles. The focus is on the correspondence between the non-rigorous replica analysis of statistical mechanics and the performance assessment of VAMP in the model-mismatched setting. The correspondence of this kind is well-known when the measurement matrix has independent and identically distributed entries. However, when the measurement matrix follows a general rotation-invariant matrix ensemble, the correspondence has been validated only under limited cases, such as the Bayes optimal inference or the convex empirical risk minimization. The result presented in this paper is to extend the scope of such correspondence. Herein, we heuristically derive the explicit formula of state-evolution equations, which macroscopically describe VAMP dynamics for the current model-mismatched case, and show that their fixed point is generally consistent with the replica symmetric solution obtained by the replica method of statistical mechanics. We also show that the fixed point of VAMP can exhibit a microscopic instability, which indicates that message variables continue to move by VAMP while their macroscopically summarized quantities converge to fixed values. The critical condition the for microscopic instability agrees with that for breaking the replica symmetry that is derived within the non-rigorous replica analysis. The results of the numerical experiments cross-check our findings.

Energy spectrum of interacting gas: Cluster expansion method

In this paper, we calculate the energy spectrum of interacting gases by converting the cluster expansion method in statistical mechanics into a method of solving energy eigenvalues. We obtain an explicit expression of the energy eigenvalue, by which we can calculate the eigenvalue of an interacting gas from the interparticle potential directly. As an example, we calculate the energy spectrum for an interacting gas with soft-sphere potentials.

Understanding probability and irreversibility in the Mori-Zwanzig projection operator formalism

Explaining the emergence of stochastic irreversible macroscopic dynamics from time-reversible deterministic microscopic dynamics is one of the key problems in philosophy of physics. The Mori-Zwanzig (MZ) projection operator formalism, which is one of the most important methods of modern nonequilibrium statistical mechanics, allows for a systematic derivation of irreversible transport equations from reversible microdynamics and thus provides a useful framework for understanding this issue. However, discussions of the MZ formalism in philosophy of physics tend to focus on simple variants rather than on the more sophisticated ones used in modern physical research. In this work, I will close this gap by studying the problems of probability and irreversibility using the example of Grabert’s time-dependent projection operator formalism. This allows to better understand how general proposals for understanding probability in statistical mechanics, namely (a) quantum approaches and (b) almost-objective probabilities, can be accomodated in the MZ formalism. Moreover, I will provide a detailed physical analysis, based on the MZ formalism, of various proposals from the philosophical literature, such as (a) Robertson’s theory of justifying coarse-graining via autonomous macrodynamics, (b) Myrvold’s problem of explaining autonomous macrodynamics, and (c) Wallace’s simple dynamical conjecture.

Theoretical prediction of melting curves of gold and silver up to pressure 150 GPa

We investigate melting curves of gold and silver thanks to the statistical moment method in quantum statistical mechanics and the modified Lindemann melting criterion. Numerical calculations have been performed for face-centered cubic (fcc) gold and silver metals up to pressure 150 GPa. Our results are compared with available experimental data and ab initio molecular dynamics simulations where possible. We show that theoretical melting line of gold goes along with recent X-ray diffraction measurements reported by Weck et al. up to pressure more than 110GPa. For silver, our work virtually coincides with the quantum molecular dynamics melting curve which is a combination of fcc and body-centered cubic (bcc) segments freshly implemented by Baty et al. At the pressure 94.2 GPa of the predicted fcc–bcc-liquid triple point of silver, our calculated melting temperature is 4302 K that is only about 2% difference from the prediction of Baty et al. This research proposes a relatively simple approach made from the modified Lindemann criterion and the moment method for predicting melting points of materials at high pressure.